Step |
Hyp |
Ref |
Expression |
1 |
|
segcon2 |
|- ( ( N e. NN /\ ( A e. ( EE ` N ) /\ B e. ( EE ` N ) ) /\ ( C e. ( EE ` N ) /\ D e. ( EE ` N ) ) ) -> E. x e. ( EE ` N ) ( ( B Btwn <. A , x >. \/ x Btwn <. A , B >. ) /\ <. A , x >. Cgr <. C , D >. ) ) |
2 |
|
andir |
|- ( ( ( B Btwn <. A , x >. \/ x Btwn <. A , B >. ) /\ <. A , x >. Cgr <. C , D >. ) <-> ( ( B Btwn <. A , x >. /\ <. A , x >. Cgr <. C , D >. ) \/ ( x Btwn <. A , B >. /\ <. A , x >. Cgr <. C , D >. ) ) ) |
3 |
|
simpl1 |
|- ( ( ( N e. NN /\ ( A e. ( EE ` N ) /\ B e. ( EE ` N ) ) /\ ( C e. ( EE ` N ) /\ D e. ( EE ` N ) ) ) /\ x e. ( EE ` N ) ) -> N e. NN ) |
4 |
|
simpl2l |
|- ( ( ( N e. NN /\ ( A e. ( EE ` N ) /\ B e. ( EE ` N ) ) /\ ( C e. ( EE ` N ) /\ D e. ( EE ` N ) ) ) /\ x e. ( EE ` N ) ) -> A e. ( EE ` N ) ) |
5 |
|
simpr |
|- ( ( ( N e. NN /\ ( A e. ( EE ` N ) /\ B e. ( EE ` N ) ) /\ ( C e. ( EE ` N ) /\ D e. ( EE ` N ) ) ) /\ x e. ( EE ` N ) ) -> x e. ( EE ` N ) ) |
6 |
|
simpl3 |
|- ( ( ( N e. NN /\ ( A e. ( EE ` N ) /\ B e. ( EE ` N ) ) /\ ( C e. ( EE ` N ) /\ D e. ( EE ` N ) ) ) /\ x e. ( EE ` N ) ) -> ( C e. ( EE ` N ) /\ D e. ( EE ` N ) ) ) |
7 |
|
cgrcom |
|- ( ( N e. NN /\ ( A e. ( EE ` N ) /\ x e. ( EE ` N ) ) /\ ( C e. ( EE ` N ) /\ D e. ( EE ` N ) ) ) -> ( <. A , x >. Cgr <. C , D >. <-> <. C , D >. Cgr <. A , x >. ) ) |
8 |
3 4 5 6 7
|
syl121anc |
|- ( ( ( N e. NN /\ ( A e. ( EE ` N ) /\ B e. ( EE ` N ) ) /\ ( C e. ( EE ` N ) /\ D e. ( EE ` N ) ) ) /\ x e. ( EE ` N ) ) -> ( <. A , x >. Cgr <. C , D >. <-> <. C , D >. Cgr <. A , x >. ) ) |
9 |
8
|
anbi2d |
|- ( ( ( N e. NN /\ ( A e. ( EE ` N ) /\ B e. ( EE ` N ) ) /\ ( C e. ( EE ` N ) /\ D e. ( EE ` N ) ) ) /\ x e. ( EE ` N ) ) -> ( ( x Btwn <. A , B >. /\ <. A , x >. Cgr <. C , D >. ) <-> ( x Btwn <. A , B >. /\ <. C , D >. Cgr <. A , x >. ) ) ) |
10 |
9
|
orbi2d |
|- ( ( ( N e. NN /\ ( A e. ( EE ` N ) /\ B e. ( EE ` N ) ) /\ ( C e. ( EE ` N ) /\ D e. ( EE ` N ) ) ) /\ x e. ( EE ` N ) ) -> ( ( ( B Btwn <. A , x >. /\ <. A , x >. Cgr <. C , D >. ) \/ ( x Btwn <. A , B >. /\ <. A , x >. Cgr <. C , D >. ) ) <-> ( ( B Btwn <. A , x >. /\ <. A , x >. Cgr <. C , D >. ) \/ ( x Btwn <. A , B >. /\ <. C , D >. Cgr <. A , x >. ) ) ) ) |
11 |
2 10
|
syl5bb |
|- ( ( ( N e. NN /\ ( A e. ( EE ` N ) /\ B e. ( EE ` N ) ) /\ ( C e. ( EE ` N ) /\ D e. ( EE ` N ) ) ) /\ x e. ( EE ` N ) ) -> ( ( ( B Btwn <. A , x >. \/ x Btwn <. A , B >. ) /\ <. A , x >. Cgr <. C , D >. ) <-> ( ( B Btwn <. A , x >. /\ <. A , x >. Cgr <. C , D >. ) \/ ( x Btwn <. A , B >. /\ <. C , D >. Cgr <. A , x >. ) ) ) ) |
12 |
11
|
rexbidva |
|- ( ( N e. NN /\ ( A e. ( EE ` N ) /\ B e. ( EE ` N ) ) /\ ( C e. ( EE ` N ) /\ D e. ( EE ` N ) ) ) -> ( E. x e. ( EE ` N ) ( ( B Btwn <. A , x >. \/ x Btwn <. A , B >. ) /\ <. A , x >. Cgr <. C , D >. ) <-> E. x e. ( EE ` N ) ( ( B Btwn <. A , x >. /\ <. A , x >. Cgr <. C , D >. ) \/ ( x Btwn <. A , B >. /\ <. C , D >. Cgr <. A , x >. ) ) ) ) |
13 |
|
brsegle2 |
|- ( ( N e. NN /\ ( A e. ( EE ` N ) /\ B e. ( EE ` N ) ) /\ ( C e. ( EE ` N ) /\ D e. ( EE ` N ) ) ) -> ( <. A , B >. Seg<_ <. C , D >. <-> E. x e. ( EE ` N ) ( B Btwn <. A , x >. /\ <. A , x >. Cgr <. C , D >. ) ) ) |
14 |
|
brsegle |
|- ( ( N e. NN /\ ( C e. ( EE ` N ) /\ D e. ( EE ` N ) ) /\ ( A e. ( EE ` N ) /\ B e. ( EE ` N ) ) ) -> ( <. C , D >. Seg<_ <. A , B >. <-> E. x e. ( EE ` N ) ( x Btwn <. A , B >. /\ <. C , D >. Cgr <. A , x >. ) ) ) |
15 |
14
|
3com23 |
|- ( ( N e. NN /\ ( A e. ( EE ` N ) /\ B e. ( EE ` N ) ) /\ ( C e. ( EE ` N ) /\ D e. ( EE ` N ) ) ) -> ( <. C , D >. Seg<_ <. A , B >. <-> E. x e. ( EE ` N ) ( x Btwn <. A , B >. /\ <. C , D >. Cgr <. A , x >. ) ) ) |
16 |
13 15
|
orbi12d |
|- ( ( N e. NN /\ ( A e. ( EE ` N ) /\ B e. ( EE ` N ) ) /\ ( C e. ( EE ` N ) /\ D e. ( EE ` N ) ) ) -> ( ( <. A , B >. Seg<_ <. C , D >. \/ <. C , D >. Seg<_ <. A , B >. ) <-> ( E. x e. ( EE ` N ) ( B Btwn <. A , x >. /\ <. A , x >. Cgr <. C , D >. ) \/ E. x e. ( EE ` N ) ( x Btwn <. A , B >. /\ <. C , D >. Cgr <. A , x >. ) ) ) ) |
17 |
|
r19.43 |
|- ( E. x e. ( EE ` N ) ( ( B Btwn <. A , x >. /\ <. A , x >. Cgr <. C , D >. ) \/ ( x Btwn <. A , B >. /\ <. C , D >. Cgr <. A , x >. ) ) <-> ( E. x e. ( EE ` N ) ( B Btwn <. A , x >. /\ <. A , x >. Cgr <. C , D >. ) \/ E. x e. ( EE ` N ) ( x Btwn <. A , B >. /\ <. C , D >. Cgr <. A , x >. ) ) ) |
18 |
16 17
|
bitr4di |
|- ( ( N e. NN /\ ( A e. ( EE ` N ) /\ B e. ( EE ` N ) ) /\ ( C e. ( EE ` N ) /\ D e. ( EE ` N ) ) ) -> ( ( <. A , B >. Seg<_ <. C , D >. \/ <. C , D >. Seg<_ <. A , B >. ) <-> E. x e. ( EE ` N ) ( ( B Btwn <. A , x >. /\ <. A , x >. Cgr <. C , D >. ) \/ ( x Btwn <. A , B >. /\ <. C , D >. Cgr <. A , x >. ) ) ) ) |
19 |
12 18
|
bitr4d |
|- ( ( N e. NN /\ ( A e. ( EE ` N ) /\ B e. ( EE ` N ) ) /\ ( C e. ( EE ` N ) /\ D e. ( EE ` N ) ) ) -> ( E. x e. ( EE ` N ) ( ( B Btwn <. A , x >. \/ x Btwn <. A , B >. ) /\ <. A , x >. Cgr <. C , D >. ) <-> ( <. A , B >. Seg<_ <. C , D >. \/ <. C , D >. Seg<_ <. A , B >. ) ) ) |
20 |
1 19
|
mpbid |
|- ( ( N e. NN /\ ( A e. ( EE ` N ) /\ B e. ( EE ` N ) ) /\ ( C e. ( EE ` N ) /\ D e. ( EE ` N ) ) ) -> ( <. A , B >. Seg<_ <. C , D >. \/ <. C , D >. Seg<_ <. A , B >. ) ) |